Uniqueness of a positive radially symmetric solution in a ball to the Dirichlet problem for one nonlinear differential equation of the second order

In the ball $S=\{x\in R^n:|x|<1\}$ ($n\ge3$) with the boundary $\Gamma$ we consider the Dirichlet problem
\begin{gather*} \Delta u+|x|^m|u|^p=0, \quad x\in S, \\ u_\Gamma=0, \end{gather*}
where $m\ge0$, $p>1$ are constants. We prove that the problem has a unique positive radially symmetric solution.